Metrics with only finitely many isometry invariant geodesics (Q1107847)

A non-constant geodesic c: \(R\to M\) on a compact Riemannian manifold M with isometry f of finite order is f-invariant if \(c(t+1)=f(c(t))\) for all \(t\in R\). These geodesics can be described as critical points of the energy functional on the Hilbert manifold of f-invariant curves. Two f- invariant geodesics \(c_ 1\), \(c_ 2\) are geometrically equivalent if \(c_ 1(R)=c_ 2(R)\). Using the index theorem for isometry invariant geodesics by \textit{K. Grove} and \textit{M. Tanaka} [Acta Math. 140, 33-48 (1978; Zbl 0375.58010)] we generalize results of the author [On the average indices of closed geodesics (to appear in J. Differ. Geom. 28 (1988)] for closed geodesics to the case of isometry invariant geodesics. For a Riemannian metric on a compact simply-connected manifold with isometry f all of whose f-invariant geodesics are non-degenerate we show: If there are only finitely many geometrically distinct f-invariant geodesics we can express a topological invariant of M and f in terms of metric invariants of the f-invariant geodesics. As an application we obtain under certain assumptions on M and f that for a generic metric with fixed isometry f there are finitely many isometry-invariant geodesics.